Is everything relative? Length and time? Yesterday I woke up in the night after having a dream (after sleeping late watching a SciFi film on space). It had this:
Suppose you are told that the linear size of everything in the universe has been doubled overnight.
Can we test it by using the fact that the speed of light is a universal constant and has not changed?
(couldn't get any method, no idea struck me as I scanned the articles on Fizeau, Foucault and Michelson method for measuring the speed of light).
Twist:  What will happen if all the clocks in the universe start running at half the original speed in the above cases? (This thing occupied my mind for the whole day, this is definitely going to fail the methods used in the above case).
 A: Let's try to understand first the setting we are in. The sentence:

the fact that the speed of light is a universal constant and has not changed

seems to imply that the laws of physics and the universal constants they depend on remain invariant under this change of sizes. So what we are dealing with is a doubling of the distances between all physical objects without modifying the universal laws and constants of physics. 
One missing element that's very important in this question is what happens to speeds of objects after the change. As the question doesn't say anything about them, let's assume they are unchanged.
Notice that almost everything in our world has a characteristic size completely determined by universal physical laws. Consider, for example, the size of atoms: Coulomb's potential together with quantum mechanics is sufficient to specify the (size of the) region the electrons are in. Distances between atoms in a solid or a liquid are also determined this way by electromagnetic force. In smaller distances, the nuclei, and even protons and neutrons have a universal size.
So if these sizes were doubled, nothing would hold together: solids and liquids would become gases, electrons and nuclei would be free, etc. If we were the same size as before, so that we can watch this happen, we wouldn't need any measurement of the speed of light to notice it.
Perhaps a more familiar example is the trajectory of the Earth. Its current orbit satisfies that the gravitational force of the Sun acting over it is approximately the one necessary for circular motion. If the distance is doubled, this force is divided by four, and with the same velocity, the earth woudn't stay in its circular motion around the Sun.
Now, to answer the question about the test with the speed of light:

If somehow you manage to do a measurement under this conditions of the time light takes to go from one object to another whose distance previous to the change you know, the result will be that light takes now the double of the time it took before, and thus you will know that distances have doubled.


About the last question involving time, I understand you are not referring just to the speed of clocks but to the speed of everything. 
If the speed of every object is halved at the same time distances are doubled, the scenario will change a bit with respect to the one in which just distances were doubled, but the main qualitative conclusions will be the same. Observe, for example, that half the speed is not enough for the Earth to stay in circular motion; instead, $1/\sqrt{2}$ of the speed is needed.
A: Funny game! From "the linear size of everything in the universe has been doubled overnight", I can first deduce that the distance in meters that light travels in free space in one second (one "thing" among "everything") has been doubled. Consequence: the speed of light has been doubled. This is in contradiction with your second statement "the speed of light is a universal constant and has not changed". Answer needs to stop here because of glaring contradiction in the question!
A: To answer the question without the twist, think about what the speed of light is in terms of its most fundamental properties:
\begin{align}
x=ct.
\end{align}
That is, the speed of light has a characteristic distance per unit time. Now in universe (1) (before the lengthening of everything), suppose that a given rod has length $x$. Then, it would take light a time $t$ to travel a distance $x$ (i.e. the length of the rod). Now consider universe (2); we must measure how long it takes for light to travel the length of the rod. Let $t'$ be the time it takes to travel the length of the rod in universe (2) and $x'$ be the length of the rod in universe (2). If $t'>t$ then $ct'>ct$ and hence $x'>x$. Therefore, you must conclude that the rod in universe (2) is longer.
This question is actually equivalent to the following question: 
I lock you in room with nothing but a rod and the constancy of the speed of light (and of course a way of measuring how long it takes for light to get from one point to another). Without your knowing, I am able to switch out the rod for another rod twice as long. Will you ever be able to tell if I switched out the rod or not? Of course you will. If it takes light longer to travel from the beginning to the end of a given rod, then the rod must be longer since the speed of light is the same. 
A: No. Our limited often literalist thinking makes us want to believe there are no exceptions to that which we have been told and/or proven. The difficulty is that Quantum Entanglement (QE) is an obvious exception inasmuch as it is faster than the speed of light. So while the standard of the speed of light is a real number the speed of QE doesn't fit with the standard and so cannot be relative in length or time. 
A: Suppose one fine morning you are told that the linear length of everything in the universe has doubled, this only means that you are on a ship that started moving with uniform velocity while you were asleep (which is to say that you did not feel the initial kick, and have good faith in the special theory of relativity) last night! 
Now by Lorentz transformation relations you can easily conclude that you are moving at approximately a speed of abut 0.866 times the speed of light.
In your second case if you assume that you are moving on some ship and the clock speed slows by a factor of half, you will on applying the Lorentz transformation relations see that your velocity becomes imaginary, which is clearly not physical. Thus, your second situation is not possible!
Note that here I assumed that by everything you mean all of the observable universe except the spaceship that you are currently travelling in!
Hope this helps.
